{"title":"On the structure of certain $\\Gamma$-difference modules","authors":"E. D. Shalit, J. Guti'errez","doi":"10.4171/lem/1032","DOIUrl":null,"url":null,"abstract":"This is a largely expository paper, providing a self-contained account on the results of [Sch-Si1, Sch-Si2], in the cases denoted there 2Q and 2M. These papers of Schäfke and Singer supplied new proofs to the main theorems of [Bez-Bou, Ad-Be], on the rationality of power series satisfying a pair of independent q-difference, or Mahler, equations. We emphasize the language of Γ-difference modules, instead of difference equations or systems. Although in the two cases mentioned above this is only a semantic change, we also treat a new case, which may be labeled 1M1Q. Here the group Γ is generalized dihedral rather than abelian, and the language of equations is inadequate. In the last section we explain how to generalize the main theorems in case 2Q to finite characteristic.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"28 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"L’Enseignement Mathématique","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4171/lem/1032","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
This is a largely expository paper, providing a self-contained account on the results of [Sch-Si1, Sch-Si2], in the cases denoted there 2Q and 2M. These papers of Schäfke and Singer supplied new proofs to the main theorems of [Bez-Bou, Ad-Be], on the rationality of power series satisfying a pair of independent q-difference, or Mahler, equations. We emphasize the language of Γ-difference modules, instead of difference equations or systems. Although in the two cases mentioned above this is only a semantic change, we also treat a new case, which may be labeled 1M1Q. Here the group Γ is generalized dihedral rather than abelian, and the language of equations is inadequate. In the last section we explain how to generalize the main theorems in case 2Q to finite characteristic.